tailieunhanh - Essentials of Control Techniques and Theory_13

Tham khảo tài liệu 'essentials of control techniques and theory_13', kinh doanh - tiếp thị, quản trị kinh doanh phục vụ nhu cầu học tập, nghiên cứu và làm việc hiệu quả | 310 Essentials of Control Techniques and Theory We saw that if we followed the Maximum Principle our drive decisions rested on solving the adjoint equations p - A p For every eigenvalue of A in the stable left-hand half-plane A has one in the unstable half-plane. Solving the adjoint equations in forward time will be difficult to say the least. Methods have been suggested in which the system equations are run in forward time against a memorized adjoint trajectory and the adjoint equations are then run in reverse time against a memorized state trajectory. The boresight method allows the twin trajectories to be massaged until they eventually satisfy the boundary conditions at each end of the problem. When the cost function involves a term in u of second order or higher power there can be a solution that does not require bang-bang control. The quadratic cost function is popular in that its optimization gives a linear controller. By going back to dynamic programing we can find a solution without resorting to adjoint variables although all is still not plain sailing. Suppose we choose a cost function involving sums of squares of combinations of states added to sums of squares of mixtures of inputs. We can exploit matrix algebra to express this mess more neatly as the sum of two quadratic forms c x u x Qx u Ru. When multiplied out each term above gives the required sum of squares and cross-products. The diagonal elements of R give multiples of squares of the us while the other elements define products of pairs of inputs. Without any loss of generality Q and R can be chosen to be symmetric. A more important property we must insist on if we hope for proportional control is that R is positive definite. The implication is that any nonzero combination of us will give a positive value to the quadratic form. Its value will quadruple if the us are doubled and so the inputs are deterred from becoming excessively large. A consequence of this is that R is non-singular so